Regular Graphs with Given Girth and Restricted Circuits

Sachs, H.

doi:10.1112/jlms/s1-38.1.423

Cited by 80 publications

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“…Since any K i, j ; j ≥ i ≥ 2 contains a 4-cycle, every graph of girth 5 is K i, j -free. Sachs [27,Theorem 1] showed that there exist graphs of girth 5 and arbitrarily large degeneracy. Therefore, K i, j -free graphs are strictly more general than graphs of bounded degeneracy.…”

Section: Resultsmentioning

confidence: 99%

Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond

Philip

Raman

Sikdar

2012

ACM Trans. Algorithms

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We show that for any fixed j ≥ i ≥ 1, the k-DOMINATING SET problem restricted to graphs that do not have K i, j (the complete bipartite graph on (i + j) vertices, where the two parts have i and j vertices, respectively) as a subgraph is fixed parameter tractable (FPT) and has a polynomial kernel. We describe a polynomial-time algorithm that, given a K i, j -free graph G and a nonnegative integer k, constructs a graph H (the "kernel") and an integer k ′ such that (1) G has a dominating set of size at most k if and only if H has a dominating set of size at most k ′ , (2) H has The most general class of graphs for which a polynomial kernel was previously known for k-DOMINATING SET is the class of K h -topologicalminor-free graphs [21]. Graphs of bounded degeneracy are the most general class of graphs for which an FPT algorithm was previously known for this problem. K h -topological-minor-free graphs are K h,h -free (but not vice versa), and so our results show that k-DOMINATING SET has both FPT algorithms and polynomial kernels in more general classes of graphs.Using the same techniques, we also obtain an O jk i vertex-kernel for the k-INDEPENDENT DOMINATING SET problem on K i, j -free graphs.

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Section: Resultsmentioning

confidence: 99%

Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond

Philip

Raman

Sikdar

2012

ACM Trans. Algorithms

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“…Their results can be viewed as biregular versions of the results from [37] and [12]. The paper [16] contains two constructions: a recursive one and an algebraic one.…”

Section: 2mentioning

confidence: 99%

“…This was first accomplished by Sachs, who in [37] showed by explicit construction that (k, g)-graphs of finite order exist. In the same year, Erdős and Sachs [12] gave, without explicit construction, a much smaller general upper bound on ν(k, g).…”

Section: 4mentioning

confidence: 99%

“…inserting a new vertex on) each edge of the original graph. The methods of [37] and [12] differ in spirit. In the paper of Sachs, the graphs are constructed explicitly but are rather sparse in their number of edges.…”

Section: 2mentioning

confidence: 99%

“…Trivially, (r, s, 2)-graphs exist for all r, s ≥ 2; indeed, these are the complete bipartite graphs. For all r, t ≥ 2, Sachs [37], and Erdős and Sachs [12], constructed r-regular graphs with girth 2t. From such graphs, (r, 2, t)-graphs can be trivially obtained by subdividing (i.e.…”

Section: 2mentioning

confidence: 99%

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General properties of some families of graphs defined by systems of equations

Lazebnik

Woldar

2001

Journal of Graph Theory

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Abstract. In this paper we present a simple method for constructing infinite families of graphs defined by a class of systems of equations over commutative rings. We show that the graphs in all such families possess some general properties including regularity and bi-regularity, existence of special vertex colorings, and existence of covering maps -hence, embedded spectra -between every two members of the same family. Another general property, recently discovered, is that nearly every graph constructed in this manner edge-decomposes either the complete, or complete bipartite, graph which it spans.In many instances, specializations of these constructions have proved useful in various graph theory problems, but especially in many extremal problems. A short survey of the related results is included. We also show that the edgedecomposition property allows one to improve existing lower bounds for some multicolor Ramsey numbers.

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Biregular Cages of Odd Girth

Exoo

Jajcay

2015

J. Graph Theory

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Abstact:Biregular ({r, m}; g)-cages are graphs of girth g that contain vertices of degrees r and m and are of the smallest order among all such graphs. We show that for every r ≥ 3 and every odd g = 2t + 1 ≥ 3, there exists an integer m 0 such that for every even m ≥ m 0 , the biregular ({r, m}, g)-cage is of order equal to a natural lower bound analogous to the well-known Moore bound. In addition, when r is odd, the restriction on the parity of m can be removed, and there exists an integer m 0 such that a biregular ({r, m}, g)-cage of order equal to this lower bound exists for all m ≥ m 0 . This is in stark contrast to the result classifying all cages of degree k and girth g whose order is equal to the Moore bound. C 2015 Wiley Periodicals, Inc. J. Graph Theory 81: 50-56, 2016Lemma 1.1. The only ({r, m}; g)-cages of even girth g whose order is equal to (1) are the ({2, m}; g)-cages and the ({r, m}; 4)-cages.Proof. The fact that the order of ({2, m}; g)-cages and of ({r, m}; 4)-cages is equal to (1) was proved in [4]. The necessity portion of the lemma follows from [12] which contains a stronger lower bound than (1) for the order of the ({r, m}, 6)-cages and from [1] which contains a stronger lower bound on the order of the ({r, m}, 2k)-cages, k ≥ 4.Except for the biregular cages covered by Lemma 1.1, all known results suggest the existence of significantly many parameter sets ({r, m}; g) for which there exists a ({r, m}; g)-graph of order equal to the lower bound (1) -a dramatic difference when compared to the case of cages. In our article, we confirm the above observation by Journal of Graph Theory

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Regular Graphs with Given Girth and Restricted Circuits

Cited by 80 publications

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Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond

Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond

General properties of some families of graphs defined by systems of equations

Biregular Cages of Odd Girth

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